When the Khanhnion Commutator Becomes a Lie Bracket: Associative Internal Products, Cross-Type Brackets, and a Sufficient Condition for the Jacobi Identity

This paper identifies a sufficient condition under which the commutator of the 128-dimensional Khanhnion scalar–vector extension defines a Lie bracket on its 96-dimensional vector sector. Paper 7 introduced the algebra A = (R ⊕ R³) ⊗ W, where W is a 32-dimensional internal algebra. For a commutative internal product β, the commutator of two elements has no scalar component and lies in the vector sector: [A, B]⊙ = (0, 2Cβ(x, y)). Paper 9 showed that, for componentwise symmetrized sedenion multiplication, the induced bracket does not satisfy the Jacobi identity. This paper proves the complementary result: if the internal product β is commutative and associative, then the bracket {x, y} = 2Cβ(x, y) satisfies the Jacobi identity. Therefore, it is a Lie bracket on the 96-dimensional vector sector. Here (Cβ(x, y))k = Σ εijk β(xi, yj), where εijk is the Levi-Civita symbol. The proof reduces the Jacobi expression to the classical vector triple-product identity. Associativity handles the internal parentheses, while commutativity ensures the required cancellations. A concrete example is obtained by taking W = R³² with componentwise multiplication. This internal product is commutative and associative. The resulting 128-dimensional algebra is unital and rotation-equivariant, and its vector-sector commutator is a Lie bracket. The paper emphasizes that only the commutator restricts to a Lie bracket. The full 128-dimensional product is not anticommutative, so the full algebra itself is not a Lie algebra. The result contrasts with Paper 9: - Symmetrized sedenion multiplication is commutative but nonassociative; Jacobi fails.- Componentwise multiplication on R³² is commutative and associative; Jacobi holds. No claim is made that commutativity and associativity are necessary for Jacobi. Nor is it claimed that the construction gives a new Lie algebra unrelated to the cross product. No new sphere-packing density bound, proof of the Kepler theorem, physical quantum model, gauge symmetry, conservation law, or computational advantage is established. Author: Nguyen Kim KhanhProject: Khanh MathematicsDate: 29 October 2026Series: Paper 10 AI assistance was used in mathematical formulation, organization, and wording. Independent verification of the arguments, definitions, and relationship to prior work is required before publication.